
The thin-shell estimate needed for Theorem REF  follows from a recent result due to Chen [1]}: setting \(\theta _m=\sqrt{\log {m} \log \log {m}}\) , Chen's estimate on the thin-shell parameter implies that there is an absolute constant \(c\)  such that for \(u \ge 1\) , with probability at least \(1-2\exp (-u)\) ,
\(\left| \frac{\Vert X\Vert _2^2}{m}-1\right| \le cu\frac{\theta _m}{\sqrt{m}}.\) 
Therefore, \(X\)  satisfies the wanted thin-shell bound with probability \(1-\gamma \)  for
\(\delta =c \frac{ \theta _m }{\sqrt{m}} \cdot \log \left(\frac{en}{\gamma }\right).\) 

By Borell's Lemma (see, e.g., [2]}), a log-concave random vector satisfies a \(\psi _1\) -\(L_1\)  norm equivalence; that is, for every \(x \in \mathbb {R}^m\)  and every \(p \ge 1\) , \(\Vert \left\langle X,x \right\rangle \Vert _{L_p} \le C p \Vert \left\langle X,x \right\rangle \Vert _{L_1}\) .

